Theorem tngip

Description: The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypotheses	tngbas.t	|- T = ( G toNrmGrp N )
	tngip.2	|- ., = ( .i ` G )
Assertion	tngip	|- ( N e. V -> ., = ( .i ` T ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	|- T = ( G toNrmGrp N )
2		tngip.2	|- ., = ( .i ` G )
3		ipid	|- .i = Slot ( .i ` ndx )
4		slotstnscsi	|- ( ( TopSet ` ndx ) =/= ( Scalar ` ndx ) /\ ( TopSet ` ndx ) =/= ( .s ` ndx ) /\ ( TopSet ` ndx ) =/= ( .i ` ndx ) )
5	4	simp3i	|- ( TopSet ` ndx ) =/= ( .i ` ndx )
6	5	necomi	|- ( .i ` ndx ) =/= ( TopSet ` ndx )
7		slotsdnscsi	|- ( ( dist ` ndx ) =/= ( Scalar ` ndx ) /\ ( dist ` ndx ) =/= ( .s ` ndx ) /\ ( dist ` ndx ) =/= ( .i ` ndx ) )
8	7	simp3i	|- ( dist ` ndx ) =/= ( .i ` ndx )
9	8	necomi	|- ( .i ` ndx ) =/= ( dist ` ndx )
10	1 3 6 9	tnglem	|- ( N e. V -> ( .i ` G ) = ( .i ` T ) )
11	2 10	eqtrid	|- ( N e. V -> ., = ( .i ` T ) )